Part B - Calculate the torque on the Yoyo about the center of mass, and the net horizontal force.

Torque = r x F = rF since r is perp to F
Fnet x = F cos θ

Part C - What is the angular accel. of the yoyo about its center of mass? Linear acceleration?

Torque = rF = Iα α = rF/I α = 2rF/MR2

and since alin = αr
a = 2r2F/MR2

Part D - Find the net torque about the instantaneous point of contact

Ok heres where I get stuck. Torque = rF sin σ , where σ is the angle between the applied force F and the r vector, right? Well no joke the geometry is complex. I cant find any way to get the angle between these two, here's what I've got so far:

σ is the angle between the two blue lines, which represent the force and the r vector. Ive also extended the force line so you can see the line of action, if it helps.

Staff: Mentor

The picture you see above is the bottom half of a YoYo of mass M. The inner axle has a radius r and the outer radius is R. The string is wrapped under the axle and being pulled gently so the YoYo rolls to the left without slipping.

Im asked to find the net Torque on the YoYo about the instantaneous point of contact with the surface. Trouble is I cant find the angle between the vectors, which is what the above diagram is trying to accomplish.